Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

Sesion Especial 16

Integracion temporal de ecuaciones diferenciales

Organizadores• Severiano Gonzalez-Pinto (Universidad de La Laguna)• Inmaculada Higueras (Universidad Publica de Navarra)• Juan Ignacio Montijano (Universidad de Zaragoza)

DescripcionEl objetivo de la sesion especial es mostrar los ultimos avances en metodos numericos para laintegracion temporal de ODEs y PDEs en el contexto del metodo de lıneas.

Programa

Martes, 5 de febrero (manana)

11:30 – 12:00 Ernst Hairer (Universite de Geneve)Long-time behaviour of numerical integrators forcharged particle dynamics

12:00 – 12:30 Luigi Brugnano (Universita degli Studi di Firenze)Spectral Hamiltonian Boundary Value Methods (SHB-VMs)

12:30 – 13:00 Pep Mulet Mestre (Universitat de Valencia)Implicit-explicit schemes for partial differential equa-tions with a nonlocal gradient flow structure

13:00 – 13:30 Laura Portero (Universidad Publica de Navarra)Parallel space-time algorithms for the solution ofparabolic problems

Martes, 5 de febrero (tarde)

17:00 – 17:30 Inmaculada Higueras (Universidad Publica de Navarra)Optimal monotonicity–preserving perturbations of agiven Runge-Kutta method

17:30 – 18:00 Domingo Hernandez Abreu (Universidad de La Laguna)On step-size control for Radau IIA methods by means oftwo-step error estimators

18:00 – 18:30 Teo Roldan (Universidad Publica de Navarra)On Low-Storage SSP(5,3) Runge-Kutta methods

18:30 – 19:00 Luis Randez (Universidad de Zaragoza)L-stable singly implicit Peer methods for the solution ofstiff IVPs

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

Long-time behaviour of numerical integrators for charged particle dynamics

Ernst Hairer

Universite de Geneve

Ernst.Hairer@unige.ch

Abstract. The Boris algorithm is the most popular time integrator for charged par-ticle motion in electric and magnetic force fields. It is a symmetric one-step method,and it preserves the phase volume exactly. However, it is not symplectic.In this talk we prove near-conservation of energy over very long times in the specialcases where either the magnetic field is constant or the electric potential is quadratic.When none of these assumptions is satisfied, it is illustrated by numerical examplesthat the numerical energy can have a linear drift or its error can behave like a randomwalk.We thank Martin Gander for drawing our attention to this problem.

References

[1] E. Hairer and C. Lubich, Energy behaviour of the Boris method for charged-particle dynamics,BIT, to appear.

[2] E. Hairer and C. Lubich, Long-term analysis of a variational integrator for charged-particle dy-namics in a strong magnetic field, Submitted for publication.

This talk is based on joint-work with Christian Lubich.

This work is partially supported by MINECO project MTM2016-77735-C3-3-P.

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Spectral Hamiltonian Boundary Value Methods (SHBVMs)

Luigi Brugnano

Universita degli Studi di Firenze

luigi.brugnano@unifi.it

Abstract. Hamiltonian Boundary Value Methods (HBVMs) [1] can be regardedas spectral methods in time along the orthonormal Legendre basis. Their usage insuch a fashion has begun only recently, by using the methods for solving highly-oscillatory problems [4] and stiffly-oscillatory problems [2], which has led to defineSpectral HBVMs (SHBVMs). This has been possible because of the very efficientnonlinear iteration available for solving the generated discrete problems [3]. In thistalk, we report the main facts concerning this approach, as well as recent findings inSHBVMs.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

References

[1] L. Brugnano, F. Iavernaro, Line Integral Methods for Conservative Problems. Chapman andHall/CRC, Boca Raton, FL, 2016.

[2] L. Brugnano, F. Iavernaro, J.I. Montijano, L. Randez, Spectrally accurate space-time solution ofHamiltonian PDEs. (2018) https://doi.org/10.1007/s11075-018-0586-z

[3] L. Brugnano, F. Iavernaro, D. Trigiante, A note on the efficient implementation of HamiltonianBVMs. J. Comput. Appl. Math. 236 (2011) 375–383.

[4] L. Brugnano, J.I. Montijano, L. Randez, On the effectiveness of spectral methods for the numericalsolution of multi-frequency highly-oscillatory Hamiltonian problems. Numer. Algorithms (2018)http://dx.doi.org/10.1007/s11075-018-0552-9

This work was partially supported by MINECO project MTM2016-77735-C3-1-P.

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Implicit-explicit schemes for partial differential equations with a nonlocalgradient flow structure

Pep Mulet Mestre

Universitat de Valencia

pep.mulet@uv.es

Abstract. The work in [1] is concerned with numerical methods for a nonlinearnonlocal equations with a gradient flow structure that arises in models for collec-tive behavior. The numerical solution of this problem by an explicit finite differencemethod is costly due to the spatial convolution in the convective numerical flux, anddue to the disadvantageous CFL condition incurred by the diffusion term. Based onexplicit schemes for such models devised in [2] a second-order implicit-explicit Runge-Kutta (IMEX-RK) method can be formulated. This method avoids the restrictivetime step limitation of explicit schemes since the diffusion term is handled implicitly.Although this entails solving nonlinear algebraic systems in every time step, numericalexperiments illustrate the relative efficiency of this proposal.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

References

[1] R. Burger, D. Inzunza, P. Mulet, Implicit-explicit schemes for nonlinear nonlocal equations witha gradient flow structure in one space dimension, to appear in Numerical Methods in PDE, 2018.

[2] J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equationswith a gradient flow structure, Commun. Comput. Phys. vol. 17 (2015) pp. 233–258.

Joint work with Raimund Burger, Daniel Inzunza (U. Concepcion, Chile) y Luis Miguel Villada (U.

Bio Bio, Chile).

Financiado por proyecto MINECO MTM2017-83942-P.

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Parallel space-time algorithms for the solution of parabolic problems

Laura Portero

Universidad Publica de Navarra

laura.portero@unavarra.es

Abstract. Classical domain decomposition techniques provide efficient numericalstrategies for the parallel solution of elliptic problems. Nevertheless, the use of suchtechniques in the context of parabolic problems encounters a bottleneck due to sequen-tial time integration procedures. Since the development of new computer architecturesis moving towards increasing the number of cores rather than improving their clockspeed-up, it is convenient to look for numerical methods that allow for time par-allelization. In this framework, this work proposes and analyzes the combination ofdomain decomposition splitting time integrators with parallel-in-time techniques suchas parareal and waveform relaxation methods (cf. [1, 2, 3] and references therein).

References

[1] M.J. Gander, 50 years of time parallel time integration, Multiple shooting and time domain de-composition methods, Contrib. Math. Comput. Sci. 9, pp. 69–113, Springer, Cham, 2015.

[2] M.J. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method,SIAM J. Sci. Comput. 29, (2007), 556–578.

[3] S. Vandewalle, Parallel multigrid waveform relaxation for parabolic problems, Teubner Skr. Nu-mer., B.G. Teubner, Stuttgart, 1993.

Joint work with Andres Arraras, Francisco J. Gaspar and Carmen Rodrigo.

This work is partially supported by the FEDER/MINECO project MTM2016-75139-R.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

Optimal monotonicity–preserving perturbations of a given Runge-Kutta method

Inmaculada Higueras

Universidad Publica de Navarra

higueras@unavarra.es

Abstract. Perturbed Runge–Kutta methods (also referred to as downwind RKmethods) can guarantee monotonicity preservation under larger step sizes relative totheir traditional Runge–Kutta counterparts. In this talk we we show how to optimallyperturb a given method in order to increase the radius of a.m. We prove that formethods with zero radius of a.m., it is always possible to give a perturbation withpositive radius. We first study methods for linear problems and then methods fornonlinear problems. In each case, we prove upper bounds on the radius of a.m.and provide algorithms to compute optimal perturbations. Optimal perturbations formany known methods are given. The numerical experiments done prove the sharpnessof the new step size restrictions [1].

References

[1] I. Higueras, D. Ketcheson, and T.A. Kocsis, Optimal monotonicity–preserving perturbations of agiven Runge-Kutta method. J. Sci. Comput, 76 (3):1337–1369, 2018.

Joint work with D. Ketcheson and T. A. Kocsis.

This work is supported by MINECO project MTM2016-77735-C3-2-P.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

On step-size control for Radau IIA methods by means of two-step errorestimators

Domingo Hernandez Abreu

Universidad de La Laguna

dhabreu@ull.edu.es

Abstract. We shall discuss error estimators for s−stage RadauIIA methods basedon two consecutive steps through embedded methods of the form

yn+1 = yn + hn−1

s∑i=1

dn,if(Yn−1,i) + hn

s∑i=1

bn,if(Yn,i),

where Yn−j,isi=1 are the internal stages computed at time tn−j, j = 0, 1, and

dn,i, bn,isi=1 are certain coefficients to be conveniently selected. Embedded meth-ods of order s with damping for the stiff components regardless of the step-size ratiohn/hn−1 are provided for 2 ≤ s ≤ 7. Numerical experiments will be presented toillustrate the performance of the newly proposed error estimator for the case s = 3incorporated in the RADAU5 code.The authors are very grateful to Professor Ernst Hairer for his consent to use andmodify the code RADAU5..

References

[1] S. Gonzalez-Pinto, J.I. Montijano, S. Perez-Rodrıguez, Two-Step Error Estimators For ImplicitRunge-Kutta Methods Applied to Stiff Systems, ACM Trans. Math. Soft., 30 no.1 (2004) 1-18.

[2] E. Hairer, G.Wanner, Solving Ordinary Differential Equations II, Stiff and Differential AlgebraicProblems, Springer 2nd ed., Berlin, 1996.

Joint work with S. Gonzalez-Pinto and J.I. Montijano.

This work was supported by MINECO project MTM2016-77735-C3-3-P.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

On Low-Storage SSP(5,3) Runge-Kutta methods

Teo Roldan

Universidad Publica de Navarra

teo@unavarra.es

Abstract. Time stepping Strong Stability Preserving (SSP) methods preserve, undercertain step size restrictions, strong stability properties of the numerical solution. Onthe other hand, for high dimension ODEs, the computer memory capacity may becompromised and low memory usage strategies must be incorporated. In the context ofSSP explicit Runge–Kutta schemes, some low-storage methods have been consideredin the literature.In this talk we study the family of third order 5-stage SSP explicit Runge–Kutta meth-ods (SSP(5,3)). First, we analyze some properties of the optimal schemes, obtainingthe best optimal SSP(5,3) methods. However, as any of them can be implementedin 2N memory registers, the study is extended to non-optimal SSP(5,3) methods. Inthis way, we achieve 2N low-storage SSP(5,3) schemes with SSP coefficients close tothe optimal one and some other relevant properties. [1]

References

[1] I. Higueras and T. Roldan, New third order low-storage SSP explicit Runge-Kutta methods, 2018;arXiv:1809.04807.

Joint work with I. Higueras

This work is supported by MINECO project MTM2016-77735-C3-2-P.

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Congreso Bienal de la Real Sociedad Matematica EspanolaSantander, 4 - 8 febrero 2019

L-stable singly implicit Peer methods for the solution of stiff IVPs

Luis Randez

Universidad de Zaragoza

randez@unizar.es

Abstract. In this paper a one parameter family of s-stage singly implicit two-steppeer (SIP) methods with order (s− 1) that are L-stable has been developed. Generalpeer methods are multistage two-step methods for solving IVPs where all stages pos-sess essentially the same accuracy and stability properties. In particular a s-stage SIPrequires at each step the solution of s-implicit non-linear systems of equations of thesame type in a similar way to the singly implicit Runge-Kutta methods. Here for each(s ≥ 3) a family of one parameter s-stage SIP methods with order (s− 1) is derived.For s ≤ 8 intervals of values of the parameter that ensure L-stability are obtained.Further it is shown that under some restriction on the parameter each s-stage SIPmethod can be formulated as a cyclic multistep method of order (s− 1) and this im-plies that the Dahlquist barrier of second-order for A-stable linear multistep methodscan be broken with suitable s-stage cyclic methods of these families.

Joint work with M. Calvo, J.I. Montijano, H. Podhaisky, L. Randez.

This work was supported by MINECO project MTM2016-77735-C3-1-P.

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